Making an object fall with cell fracture
To touch on why your object your object is exploding at the beginning of the animation, check the Collision Margin in the properties panel>physics>rigid body>rigid body tools tab.If it has been changed at all, this can affect how the simulation works. 
 

If the collision margin sensitivity is ticked on and too high, (1 is the maximum) the fractures are literally pushing away from each other, thus the object explodes right away. Creating the effect you're looking for will take just a few steps:


Step one is of course to fracture your object, in my example .gif below I used a sphere.
Step two is to add rigid body physics. Delete the object on layer
one, and move the fractured object from layer two to layer one, or
just work on layer two. Select one fracture, go to the tool
panel>physics tab and click on add active.





Next, select all of the fractures and click on copy from active.





Now you need to add a collision object, so that the fractured
object has something to shatter against. Add a plane and go to the physics settings in the tool panel again, and click add passive.  




Now, drag the fractured object to the height you want it to fall
   from, and press Alt+A, and something like this
   should happen: